# Set up sharpe ratio with 2 risky portfolio

You are considering an investment in the stock. In the stock market, there are two risky stocks (A and B) and a risk free claim, C (you can think of it as the t-bill). The covariances and returns of these three stocks are described in the following table: Assume that you have a mean-variance utility function with risk aversion A=5. That is, your utility function is

Let P be the risky portfolio that consists of stocks A and B. Let Wpa be the weight of stock A in portfolio P and let Wpb = 1-Wpa be the weight of stock B in the portfolio P. Write down the Sharpe ratio of portfolio P as a function of Wpa...

I am having trouble with how to set this equation up to start, any guidance would be helpful to start this off

• Here's some guidance: what do you need the utility for? – James Oct 10 '14 at 2:24

Sharpe Ratio is defined as the slope of the line that is the return in function of the variance of a portfolio composed by a risky and a risky-less asset.

Hence, if you have a bunch of risky assets (A,B) and a risky-less (C) you simply calculate the efficient portfolio for your risky asset (A,B), and then calculate the Sharpe ratio for the tangential portfolio.

In other words the properties of the combination of A,B assets (given $x_A + x_B = 1$) are, $$\mu_e = \mu_A x_A + \mu_B x_B = \mu_A x_A + \mu_B (1-x_A)=x_A(\mu_A - \mu_B)+\mu_B,$$ $$\sigma^2_e = \sigma^2_A x^2_A + \sigma^2_B x^2_B + \sigma_{AB}x_A x_B = x^2_A (\sigma^2_A+\sigma^2_B-2\sigma_{AB}) +2 x_A(\sigma_{AB} - \sigma^2_B) + \sigma^2_B,$$ with $mu$ the expected returns, and $\sigma^2$ the risks.

This results that when you consider the risk-free asset C in the mix you have $$\mu_p = (1-x_e) r_f + x_e \mu_e = r_f + x_e(\mu_e-r_f),$$ with $r_f$ the risk-free return, in your case the 4% return of C. The variance is given only by the risky part of the portfolio, therefore $$\sigma^2_p = x^2_e \sigma^2_e,$$ from which one can conveniently define the weight to be given to the efficient portfolio as $$x_e = \sigma_p / \sigma_e,$$ determining that $$\mu_p = (1-x_e) r_f + x_e \mu_e = r_f + \frac{(\mu_e-r_f)}{\sigma_e}\sigma_p.$$

The angular coefficient of such line, telling us that one can tune the expected return on the volatility using the risk-free rate as buffer (and eventually shorting on it), is known as Sharpe Ratio, $$\frac{(\mu_e-r_f)}{\sigma_e}.$$

If you substitute $\sigma_e$ and $\mu_e$ with the above given equations, you have the sharpe ratio in function of $x_A$ (or Wpa as you call it).

For this first assignment the Utility function is not needed, but I guess it will come handy later when it will ask to use it to calculate the desired weights.

Well, you need to calculate the expected return of your portfolio, and the volatility. Taking into account that your weights are 100% long in A, 100% short in B: $$E(r)_A-E(r)_B$$ Same for the volatility of the portfolio.

Then just calculate the Sharpe ratio: $$\frac{E(r)_{portf}-E(r)_{rf}}{Volatility_{portf}}$$